Angles between lines

Geometry: Lines

Angles between lines

We have seen the slope of a line. We can use the slope to determine the angle of the line with the $x$ -axis.

Angle of inclination

The angle of inclination of a line $k$ is the acute or right angle that $k$ makes with the $x$ -axis.

A line with positive slope has a positive angle of inclination and one with a negative slope has a negative angle of inclination. Therefore we have for an angle of inclination $\blue{\alpha}$ in radians that $-\frac{\pi}{2} \lt \blue{\alpha} \leq \frac{\pi}{2}$ We have the formula

$\tan(\blue{\alpha})=\text{slope}$

Degrees

In degrees the angle of inclination of a line lies between $-90^\circ$ and $90^\circ$ .

This can be calculated with the $\arctan$ .

Using the angle of inclination of a line we can calculate the angle between two intersecting lines.

Angle between two lines

	Step-by-step	Example
	We calculate the angle between lines $\blue l$ and $\green k$ .	$\blue l: y=\sqrt{3}x+1$ $\green k: \sqrt{3}x+3y=2$
Step 1	Calculate the slopes of line $\blue l$ and line $\green k$ .	${rc}_{\blue l}=\sqrt{3}$ ${rc}_{\green{k}}=-\frac{\sqrt{3}}{3}$
Step 2	Calculate the angle of inclination $\alpha$ of the two lines.	$\alpha_{\blue{l}}=\frac{\pi}{3}$ $\alpha_{\green{k}}=-\frac{\pi}{6}$
Step 3	Calculate the angle between lines $\blue l$ and $\green k$ . If $\alpha_{\blue l} \gt \alpha_{\green{k}}$ , then: $\alpha_{\blue l}- \alpha_{\green k}$ if $\alpha_{\blue {l}}- \alpha_{\green k} \leq \frac{\pi}{2}$ $\pi-(\alpha_{\blue l}- \alpha_{\green k} )$ if $\alpha_{\blue l}- \alpha_{\green k} \gt \frac{\pi}{2}$	$\frac{\pi}{3}--\frac{\pi}{6}=\frac{\pi}{2}$

Picture

In the figure we can see the example graphed in the $xy$ -plane.

Step 3

When $\alpha_{\blue l} \lt \alpha_{\green k}$ we can interchange the roles of $\blue l$ and $\green k$ and do the same calculation.

Two Angles

Two intersecting lines always determine four angles. These angles come in opposite pairs and in these pairs the angles are equal. In general, there are two different angles. When we talk about the angle between two lines, we choose the smaller of the two. Hence we make the disctinction in step $3$ .

Determine the angle $\alpha$ of the line $y=-{{x}\over{\sqrt{3}}}-9$ .
Give an exact answer in radians.

$\alpha=$ $-{{\pi}\over{6}}$

The slope of the line $y=-{{x}\over{\sqrt{3}}}-9$ is equal to $-{{1}\over{\sqrt{3}}}$ .
Now for the angle $\alpha$ it holds that $\tan(\alpha)=-{{1}\over{\sqrt{3}}}$ .
This means that $\alpha=-{{\pi}\over{6}}$ .

New example